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Concept
Probability-generating function
Summary
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(X = i) in the probability mass function for a random variable X, and to make available the well-developed theory of power series with non-negative coefficients.
If X is a discrete random variable taking values in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as
where p is the probability mass function of X. Note that the subscripted notations GX and pX are often used to emphasize that these pertain to a particular random variable X, and to its distribution. The power series converges absolutely at least for all complex numbers z with |z| ≤ 1; in many examples the radius of convergence is larger.
If X = (X1,...,Xd ) is a discrete random variable taking values in the d-dimensional non-negative integer lattice {0,1, ...}d, then the probability generating function of X is defined as
where p is the probability mass function of X. The power series converges absolutely at least for all complex vectors z = (z1,...,zd ) ∈ Cd with max{|z1|,...,|zd |} ≤ 1.
Probability generating functions obey all the rules of power series with non-negative coefficients. In particular, G(1−) = 1, where G(1−) = limz→1G(z) from below, since the probabilities must sum to one. So the radius of convergence of any probability generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients.
The following properties allow the derivation of various basic quantities related to X:
The probability mass function of X is recovered by taking derivatives of G,
It follows from Property 1 that if random variables X and Y have probability-generating functions that are equal, , then . That is, if X and Y have identical probability-generating functions, then they have identical distributions.
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